Abstract
This is a survey on harmonic maps and biharmonic maps into (1) Riemannian manifolds of non-positive curvature, (2) compact Lie groups or (3) compact symmetric spaces, based mainly on my recent works on these topics.
Highlights
Harmonic maps play a central role in geometry; they are critical points of the energy functional RE(φ) = 12 M |dφ|2 vg for smooth maps φ of (M, g) into (N, h)
Let us recall the definition of a harmonic map φ : (M, g) → (N, h), of a compact Riemannian manifold (M, g) into another Riemannian manifold (N, h), which is an extremal of the energy functional defined by: Z
Oniciuc [20], etc.): The generalized Chen’s conjecture: A biharmonic submanifold in a Riemannian manifold of non-positive curvature must be minimal
Summary
Harmonic maps play a central role in geometry; they are critical points of the energy functional. E(φ) = 12 M |dφ|2 vg for smooth maps φ of (M, g) into (N, h). The Euler–Lagrange equations are given by the vanishing of the tension field τ (φ). Lemaire extended [1] the notion of harmonic maps to biharmonic maps, which are, by definition, critical points of the bienergy functional: Z. Notice that harmonic maps are always biharmonic by definition. The outline of this survey is the following:. (3) Outline of the proofs of Theorems 3–5. (4) Harmonic maps and biharmonic maps into compact Lie groups or compact symmetric spaces. (5) The CR analogue of harmonic maps and biharmonic maps. (6) Biharmonic hypersurfaces of compact symmetric spaces
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